CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Algebra 322 (2009) 2286–2296 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra On non-solvable Camina pairs ∗ Zvi Arad a, Avinoam Mann b, Mikhail Muzychuk a, , Cristian Pech c a Department of Computer Sciences and Mathematics, Netanya Academic College, University St. 1, 42365, Netanya, Israel b Einstein Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel c Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel article info abstract Article history: In this paper we study non-solvable and non-Frobenius Camina Received 27 September 2008 pairs (G, N). It is known [D. Chillag, A. Mann, C. Scoppola, Availableonline29July2009 Generalized Frobenius groups II, Israel J. Math. 62 (1988) 269–282] Communicated by Martin Liebeck that in this case N is a p-group. Our first result (Theorem 1.3) shows that the solvable residual of G/O (G) is isomorphic either Keywords: p e = Camina pair to SL(2, p ), p is a prime or to SL(2, 5), SL(2, 13) with p 3, or to SL(2, 5) with p 7. Our second result provides an example of a non-solvable and non- 5 ∼ Frobenius Camina pair (G, N) with |Op (G)|=5 and G/Op (G) = SL(2, 5).NotethatG has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studies by S.M. Gagola [S.M. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math. 109 (1983) 363–385]. To our knowledge this group is the first example of a Gagola group which is non-solvable and non-Frobenius. © 2009 Elsevier Inc. All rights reserved. 1. Introduction Let G be a finite group. A pair (G, N), N P G is called a Camina pair [3] if ∀g ∈ G \ N, gN ⊂ gG , (1) i.e. each element g ∈/ N is conjugate to all the elements in the coset gN.ThesubgroupN will be referred to as the Camina kernel. * Corresponding author. E-mail addresses: [email protected] (Z. Arad), [email protected] (A. Mann), [email protected] (M. Muzychuk). 0021-8693/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2009.06.023 Z. Arad et al. / Journal of Algebra 322 (2009) 2286–2296 2287 In [3] A.R. Camina proved that if (G, N) is a Camina pair then it is Frobenius group or at least one of the groups G/N, N is a p-group. D. Chillag, A. Mann and C. Scoppola proved in [6] that G is solvable if G/N is a p-group. So for a non-solvable Camina pair (G, N) the subgroup N is a p-group or G is Frobenius. All non-solvable Frobenius groups are well known (see [11]). In this paper we focus on non-solvable and non-Frobenius Camina pairs. Given a Camina pair as above, with N a p-group, let g ∈ G be a p -element, of order q,say.Ifg commutes with an element 1 = n ∈ N, where n has order pe ,thengn has order qpe = q, and thus g and gn are not conjugate. Therefore g commutes with no non-identity element in N. G/Φ(N) is also a Camina group, with an elementary abelian kernel V = N/Φ(N). In [4] P. Fleischmann, W. Lempken and P.H. Tiep say that a group G is p -semiregular on a finite- dimensional F[G] module V defined over some field F,ifeveryp -element of G acts without fixed points on the set V \{0}. Let R be the set of primes r satisfying the following conditions (a) r = 2a · 3b + 1fora 2, b 0; (b) (r + 1)/2isaprime,orr = 7or17. Theorem 4.2 of [4] states the following Theorem 1.1. Let G be a non-trivial perfect finite group with Op(G) = 1 and p ||G|,andletGactp- semiregularly on a G-module V of characteristic p. Then one of the following holds ∼ (a) G = SL(2, pa) for some a 1 and pa > 3; ∼ + (b) G = Sz(22a 1) for some a 1 and p = 2; ∼ + + (c) G = Sz(22a 1) × SL(2, 22b 1) for some a, b 1 and gcd(2a + 1, 2b + 1) = 1,andp= 2; (d) G =∼ SL(2, r) with r ∈ R and p = 3. Conversely, if (G, p) satisfies any of the conditions (a)–(d), then there exists a faithful absolutely irreducible G-module V in characteristic p such that G acts p -semiregularly on V . In view of our earlier remarks, this has the following Corollary 1.2. Let (G, N) be a Camina pair such that G is not solvable. If N is a p-group and p divides the ∞ order of (G/Op(G)) , then one of the following holds ∞ ∼ n 2m+1 (a) (G/Op (G)) = SL(2, 2 ), Sz(2 ) and p = 2; ∞ ∼ 2a+1 2b+1 (b) (G/Op (G)) = SL(2, 2 ) × Sz(2 ) with gcd(2a + 1, 2b + 1) = 1 and p = 2; ∞ ∼ n ∞ ∼ (c) (G/Op (G)) = SL(2, 3 ) or (G/Op(G)) = SL(2, r),r∈ R and p = 3; ∞ ∼ n (d) (G/Op (G)) = SL(2, p ) and p 5. ∞ Here X denotes the solvable residual of the group X, i.e. the last term in the derived series X(d). Our first result eliminates most of the cases. More precisely, we prove the following Theorem 1.3. Let (G, N) be a Camina pair such that G is non-solvable. Then N is a p-group and one of the following holds ∞ ∼ e e (a) (G/Op (G)) = SL(2, p ),p > 3; ∞ ∼ (b) (G/Op (G)) = SL(2, 5),p= 3; ∞ ∼ (c) (G/Op (G)) = SL(2, 13),p= 3; ∞ ∼ ∼ (d) (G/Op (G)) = SL(2, 5),p 7 and (S(G), N) is a Camina pair; G/S(G) = A5 or S5 (here S(G) denotes the solvable radical of G). 2288 Z. Arad et al. / Journal of Algebra 322 (2009) 2286–2296 If N is minimal, then by Proposition 2.2 the factor-group G/Op(G) acts faithfully on N by conjugation. So, N affords a natural structure of a faithful irreducible Fp[G/Op(G)]-module on ∞ which G/Op(G) acts p -semiregularly. Considering N as an Fp[(G/Op(G)) ]-module we conclude, ∞ by Clifford’s theorem, that N is isomorphic to direct sum of simple Fp[(G/Op(G)) ]-modules, say ∞ N1 ⊕···⊕ Nk. On each of these modules the group (G/Op(G)) acts p -semiregularly. All simple Fp[SL(2,)]-modules with p -semiregular action of SL(2,) were classified in [4]. This yields the fol- lowing Corollary 1.4. Each of the Ni ’s is isomorphic to e ∞ ∼ e e (a) the natural 2-dimensional module of SL(2, p ) if (G/Op (G)) = SL(2, p ),p > 3; ∞ ∼ (b) one of the two 4-dimensional F3[SL(2, 5)]-modules if (G/Op(G)) = SL(2, 5),p= 3; ∞ ∼ (c) one of the two 6-dimensional F3[SL(2, 13)]-modules if (G/Op(G)) = SL(2, 13),p= 3; ∞ ∼ (d) a 2 or 4-dimensional Fp[SL(2, 5)]-module if (G/Op (G)) = SL(2, 5),p 7. Notice that in the latter case the dimension of an Fp[SL(2, 5)]-module is two if and only if 5 and −1 are squares in Fp . Our second result provides an example of a non-solvable and non-Frobenius Camina pair (G, N) with |G|=23 · 3 · 56 (see Section 3). The group G has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studied by S.M. Gagola [5]. To our knowledge, this group is the first example of a Gagola group which is non-solvable and non-Frobenius. Arad and Blau [1] and Mann [10] studied finite groups G which have two distinct irreducible characters χ and η satisfying χη= mχ + nη with m,n > 0. (2) The interest in this situation arises because of the fact that very little is known about the decom- position into irreducibles of the product of two irreducible characters, thus there is interest even in such extreme assumptions as (2). It was shown in [2] that G is not simple. Following that, Arad and Blau [1] and Mann [10] proved the following Theorem 1.5. Let G be a finite group satisfying (2).Thenχ and η havethesamekernel,sayM,andthere exists a normal subgroup N of G such that (a) M ⊂ NandN/M is an elementary abelian minimal normal subgroup of G/M; (b) (G/M, N/M) is a Camina pair with Camina kernel N/M; (c) N/M consists of the identity and two non-trivial conjugacy classes, which are real. Conversely, a Camina pair with an elementary abelian kernel N consisting of the identity and two non-identity real classes has two faithful irreducible characters satisfying (2). As a corollary of Theorem 1.3 we obtain the following Theorem 1.6. Let G be as in Theorem 1.5. Assume that G is non-solvable and non-Frobenius. Then |N/M|=34, ∞ ∼ N ⊂ O3(G) and (G/O3(G)) = SL(2, 5). We have no example of such groups. The paper is organized as follows. The next section contains the proof of Theorems 1.3 and 1.6. Section 3 describes the example mentioned above. All the notations used in the paper are standard.